The shipping cost per component between each facility is shown in the table below.

$\backslash$table$[[,$Philadelphia,Knoxville,Memphis,New Orleans,El Paso$],[$Manhattan$,$$$3,$$$2,,,],[$Atlanta$,$$$2,$$$3,,,],[$Philadelphia$,,$$$2,$$$4,$$$4,$$$4$XYZ Clothiers designs and manufactures high$-$end garments for men. XYZ$$s facilities in Manhattan and Atlanta serve as design and component manufacturing facilities. The completed components are then shipped to warehouses in Philadelphia or Knoxville, where they are held until they are sent on to facilities for final assembly at either Memphis, New Orleans, or El Paso.

The manufacturing facilities in Manhattan and Atlanta each manufacture $900$ units. Demand in Memphis, New Orleans, and El Paso is for $450,500,$ and $620$ units, respectively. At present, the two warehouses each have sufficient capacity to store the combined output of both manufacturing facilities. Incomplete or partial components are rejected by quality control at both the manufacturing facilities and at the final assembly facilities.

The shipping cost per component between each facility is shown in the table below.

The problem is to determine the best way to ship the garment components from the component manufacturing facilities to the final assembly facilities at the lowest total cost possible.

$11$ In new lines entered below this part, define all the decision variables. Be explicit and specific $($including units$).$

$12.$ In new lines entered below this part, state mathematically the objective function in terms of the decision variables you defined in part $11.$

$13.$ In new lines entered below this part, state mathematically all the constraints of the situation described in terms of the decision variables you defined in part $11.$

$14.$ In new lines entered below this part, state whether or not the problem you formulated mathematically in parts $11$ through $13$ is a linear program $($LP$)$ or not? If it is$,$ explain why.

$15.$ In new lines entered below this part, state the specific special kind of LP problem it is$.$

$16.$ a$.$ In new lines entered below this part, state whether or not this problem balanced or unbalanced?

b$.$ If it is unbalanced, in new lines entered below this part, state which way is it unbalanced and by how much.

$17.$ Solve this problem with a computer using either Excel Solver or LINGO.

In a space created below this part, you must present HERE copies of both the computer input and output screens.

$$ AND submit all of your computer files with your completed exam to the Exam $2$ link.

$18.$ In new lines entered below this part, state all the routes that are used in your solution to ship the garment components from the component manufacturing facilities to the final assembly facilities.

$19.$ In new lines entered below this part, state the amounts shipped over each route that is used to ship the garment components from the component manufacturing facilities to the final assembly facilities.

$20.$ In new lines entered below this part, state the total cost of shipping the garment components from the component manufacturing facilities to the final assembly facilities.

$21.$ a$.$ In new lines entered below this part, state which facility or facilities got stuck with surplus components AND by how much.

b$.$ In new lines entered below this part, state which facility or facilities will not have their demand met AND by how much.

$22.$ In new lines entered below this part, state what the numerical value of each and every slack represents in the specific context of this problem.

$23.$ a$.$ In new lines entered below this part, present HERE a drawing of the initial network representation of this problem all possible the routes.

b$.$ In new lines entered below this part, present HERE a drawing of the final network representation of this problem showing only the routes that are used.

$$ On your drawing, annotate each and every route shown in your diagram with the number of components transferred and its cost.